The conjecture is known to be true in dimension two. In 1985, Balas and Gauduchon proved in [1]} that when \(c\le 0\)  the conjecture holds (see also [2]} and [3]} for earlier work). In 1996, Apostolov, Davidov and Muskarov [4]} solved the remaining \(c>0\)  case in dimension two.
